Convert 14.75 To Its Equivalent Fraction Expressed In Lowest Terms

Converting 14.75 to its Equivalent Fraction in Lowest Terms: A Comprehensive Guide

Converting decimal numbers to fractions might seem daunting at first, but with a structured approach, it becomes a straightforward process. This comprehensive guide will walk you through converting 14.75 to its equivalent fraction in the lowest terms, explaining each step in detail and providing valuable insights into the underlying mathematical concepts. We'll also explore the broader implications of this conversion and how it relates to other mathematical operations.

Understanding Decimal Numbers and Fractions

Before diving into the conversion, let's solidify our understanding of decimal numbers and fractions. A decimal number is a number expressed in the base-10 numeral system, where each digit represents a power of 10. For instance, 14.75 can be broken down as:

• 10: Represents 10¹ = 10
• 4: Represents 4 x 10⁰ = 4
• 7: Represents 7 x 10⁻¹ = 0.7
• 5: Represents 5 x 10⁻² = 0.05

A fraction, on the other hand, represents a part of a whole, expressed as a ratio of two integers: the numerator (top number) and the denominator (bottom number). For example, 1/2 represents one part out of two equal parts.

The goal of our conversion is to find a fraction that represents the same value as 14.75.

Step-by-Step Conversion of 14.75 to a Fraction

Here's a detailed step-by-step process to convert 14.75 to its equivalent fraction:

Step 1: Remove the Decimal Point

The first step involves removing the decimal point from the number 14.75. To do this, we multiply the number by a power of 10 that shifts the decimal point to the right until we have a whole number. In this case, multiplying 14.75 by 100 (10²) gives us 1475.

Step 2: Express as a Fraction

Now, we can express the resulting whole number (1475) as a fraction. Since we multiplied the original number by 100 to remove the decimal point, the denominator of our fraction will be 100. Therefore, our initial fraction is:

1475/100

Step 3: Simplify the Fraction (Finding the Lowest Terms)

The fraction 1475/100 is not in its lowest terms. To simplify it, we need to find the greatest common divisor (GCD) of the numerator (1475) and the denominator (100). The GCD is the largest number that divides both the numerator and denominator without leaving a remainder.

One way to find the GCD is through prime factorization:

• Prime factorization of 1475: 5² x 59
• Prime factorization of 100: 2² x 5²

The common factors are 5². Therefore, the GCD of 1475 and 100 is 25.

Now, we divide both the numerator and the denominator by the GCD (25):

1475 ÷ 25 = 59 100 ÷ 25 = 4

This gives us the simplified fraction:

59/4

Step 4: Verify the Conversion

To ensure accuracy, we can convert the simplified fraction back to a decimal:

59 ÷ 4 = 14.75

This confirms that our conversion is correct. The fraction 59/4 is the equivalent of 14.75 expressed in its lowest terms.

Alternative Methods for Conversion

While the method described above is the most common and straightforward, there are other approaches you can use:

Method Using the Place Value System

This method directly utilizes the place value of each digit after the decimal point:

14.75 = 14 + 0.7 + 0.05

Convert each decimal part into a fraction:

• 0.7 = 7/10
• 0.05 = 5/100

Then add the whole number part and the fractions:

14 + 7/10 + 5/100

Find a common denominator (100 in this case) and add the fractions:

14 + 70/100 + 5/100 = 14 + 75/100

Convert 14 to an improper fraction with a denominator of 100:

1400/100 + 75/100 = 1475/100

Finally, simplify this fraction as shown in the previous section to get 59/4.

Using a Calculator

While not as instructive as the manual methods, calculators can be helpful for verifying results or handling more complex decimal conversions. Many calculators have a function to convert decimals directly into fractions.

Practical Applications and Further Exploration

Converting decimals to fractions is a fundamental skill with several practical applications across various fields:

• Engineering and Construction: Precise measurements are crucial in these fields, and fractions often provide a more accurate representation than decimals.
• Cooking and Baking: Recipes frequently involve fractional measurements.
• Finance: Calculations involving percentages and interest rates often require conversions between decimals and fractions.
• Computer Science: Binary representations of numbers are often converted to fractions for easier manipulation.

Further exploration of this topic could include:

• Converting recurring decimals to fractions: Recurring decimals present a slightly more challenging conversion process.
• Converting fractions to decimals and percentages: Understanding the interconversion between these three forms is crucial for mathematical fluency.
• Advanced fraction operations: Learning to add, subtract, multiply, and divide fractions is essential for mastering this mathematical concept.

Conclusion

Converting 14.75 to its equivalent fraction in lowest terms (59/4) involves a straightforward process of removing the decimal point, expressing the resulting number as a fraction, and then simplifying the fraction by finding its greatest common divisor. Understanding this process provides a strong foundation for working with decimals and fractions in various applications. Remember to practice regularly to enhance your understanding and skills in this area of mathematics. Mastering these skills opens doors to more complex mathematical problems and helps you to confidently tackle tasks in numerous fields.